2.1 Factors, multiples and prime numbers — Learn

Prerequisites: multiplication facts and division with whole numbers.

Objectives: find factors and multiples; recognise prime and composite numbers; explain why 1 is not prime.

Covers: N4.

Factors and multiples

A factor of a number divides it exactly, with no remainder. A multiple is the result of multiplying a number by a whole number.

For 18, factor pairs are $1\times18$, $2\times9$ and $3\times6$. Therefore its factors are $1,2,3,6,9,18$.

Multiples of 6 are $6,12,18,24,30,\ldots$. The list continues by adding 6 each time.

Worked example. Find all factors of 20 and the first four positive multiples of 7.

Pair factors until the pairs reverse: $1\times20$, $2\times10$, $4\times5$.

So the factors are $1,2,4,5,10,20$.

Multiply 7 by $1,2,3,4$: $7,14,21,28$.

Prime numbers

A prime number is a whole number greater than 1 with exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite because it has one factor.

Worked example. Is 29 prime?

Check possible divisors up to 5, since $5^2<29<6^2$. Neither 2, 3 nor 5 divides 29 exactly.

No prime divisor at most $\sqrt{29}$ divides it, so 29 is prime.

Worked example. Explain why 21 is composite.

$21=3\times7$, so 3 and 7 divide it exactly.

It has factors besides 1 and itself, so it is composite.

Watch out. You may call 1 prime because it has no factor pair. A prime must have exactly two factors; 1 has only one. Check the number of factors.

Progress check

The club has 24 blue counters and 36 red counters. Which whole numbers divide both amounts? This list will help you find the largest possible pack count in the next section.

Try it: multiples on a grid

Predict first. Without using the tool, is 42 a multiple of 3? Is 7 a factor of 42?

Now pick a number and look at its multiples.

Explain: how can you tell from the grid whether one number is a factor of another?

You should now be able to

• list factors by using factor pairs;
• generate multiples in order;
• identify primes and composites using divisibility;
• explain why 1 is neither prime nor composite.